The Distances to Open Clusters from Main-Sequence Fitting. III. Improved Accuracy with Empi

a r X i v :a s t r o -p h /0607549v 2 13 J a n 2007

A STROPHYS .J.655(2007)233-260

Preprint typeset using L A T E X style emulateapj v.04/21/05

THE DISTANCES TO OPEN CLUSTERS FROM MAIN-SEQUENCE FITTING.III.

IMPROVED ACCURACY WITH EMPIRICALLY CALIBRATED ISOCHRONES

D EOKKEUN A N 1,D ONALD M.T ERNDRUP 1,M ARC H.P INSONNEAULT 1,D IAN

E B.P AULSON 2,

R OBERT B.H ANSON 3,AND J OHN R.S TAUFFER 4

Astrophys.J.655(2007)233-260

ABSTRACT

We continue our series of papers on open cluster distances with a critical assessment of the accuracy of main-sequence ?tting using isochrones that employ empirical corrections to the color-temperature relations.We use four nearby open clusters with multicolor photometry and accurate metallicities and present a new metallicity for Praesepe ([Fe /H]=+0.11±0.03)from high-resolution spectra.The internal precision of distance estimates is about a factor of 5better than the case without the color calibrations.After taking into account all major systematic errors,we obtain distances accurate to about 2%–3%when there exists a good metallicity esti-mate.Metallicities accurate to better than 0.1dex may be obtained from BVI C K s photometry alone.We also derive a helium abundance for the Pleiades of Y =0.279±0.015,which is equal within the errors to the Sun’s initial helium abundance and that of the Hyades.Our best estimates of distances are (m ?M )0=6.33±0.04,8.03±0.04,and 9.61±0.03to Praesepe,NGC 2516,and M67,respectively.Our Pleiades distance at the spectroscopic metallicity,(m ?M )0=5.66±0.01(internal)±0.05(systematic),is in excellent agreement with several geometric distance measurements.We have made calibrated isochrones for ?0.3≤[Fe /H]≤+0.2available online.

Subject headings:Hertzsprung-Russell diagram —open clusters and associations:individual (M67,

NGC 2516,Pleiades,Praesepe)—stars:distances —stars:abundances —stars:evolu-tion —stars:activity

1.INTRODUCTION

The determination of accurate distances is the key to under-standing how stars and the Galaxy have formed and evolved.From protostars in star-forming regions to ancient tracers of the halo,improved distances have re?ned stellar evolution-ary theory and Galactic structure models (e.g.,Reid 1999).The Hipparcos mission (Perryman et al.1997)was especially valuable,providing trigonometric parallaxes for ～105stars to precision of 1–2mas (Perryman et al.1997).These par-allaxes,however,are only useful for individual stars within ～100pc.Most open clusters are much more distant than this “horizon,”but a half-dozen of the nearest clusters have 10–50or more Hipparcos stars,yielding cluster parallaxes ostensi-bly accurate to 5%or better (Mermilliod et al.1997;Perryman et al.1998;Robichon et al.1999;van Leeuwen 1999).

Main-sequence (MS)?tting,also known as the photomet-ric parallax method (e.g.,Johnson 1957;Siegel et al.2002),has long been used to estimate distances to individual stars and star clusters beyond the limits of parallax studies,and is considered to be a robust and well-understood technique.It was therefore a big surprise when the Hipparcos distances to the Pleiades and Coma Ber open clusters were in disagree-ment with distances from the MS-?tting at more than a 3σlevel (Pinsonneault et al.1998).It is dif?cult to reconcile a short Pleiades distance with stellar interior and spectroscopic abundance studies.A high helium abundance would make

1

Department of Astronomy,Ohio State University,140West 18th Av-enue,Columbus,OH 43210;deokkeun,terndrup,pinsono@https://www.doczj.com/doc/9017065911.html,.

2Planetary Systems Branch,Code 693,NASA Goddard Space Flight Cen-ter,Greenbelt,MD 20771;diane.b.paulson@https://www.doczj.com/doc/9017065911.html,.

3University of California Observatories/Lick Observatory,Santa Cruz,CA 95064;hanson@https://www.doczj.com/doc/9017065911.html,.

4Infrared Processing and Analysis Center,California Institute of Technol-ogy,Pasadena,CA 91125;stauffer@https://www.doczj.com/doc/9017065911.html,.

a cluster fainter than expected from its metallicity,and solu-tions of this type have been discussed in the literature for the Pleiades (Belikov et al.1998).However,this is dif?cult to understand since there do not seem to be nearby ?eld stars of similar characteristics in the Hipparcos catalog (Soderblom et al.1998),and the helium enhancement would have to be enormous (Y ≈0.34).In addition,it has been suggested that the metal abundance from spectroscopy may have been sig-ni?cantly overestimated (Percival et al.2003).An argument was also made that distance estimates from theoretical stel-lar models have been overestimated for young clusters due to unknown,age-related physics (van Leeuwen 1999).

However,the most likely explanation is related to the Hip-parcos parallaxes themselves.Pinsonneault et al.(1998)showed that the 12bright stars near the center of the Pleiades all had virtually the same parallax,～9mas,more than 1mas larger than the mean parallax for other cluster stars.They at-tributed this to a local zero-point error of the individual stellar

parallaxes that are correlated over the Hipparcos ’0.?

9?eld of view (van Leeuwen &Evans 1998).These quasi-random errors were caused by the Hipparcos great-circle data reduc-tions,as Makarov (2002,2003)proved by re-reducing the Pleiades and Comar Ber cluster parallaxes in a different way that correctly obtains the absolute zero point of parallax.Ad-ditional effects may result from the way the Hipparcos data were obtained and analyzed,and a more elaborate reduction of the Hipparcos parallaxes promises to produce improved distances and better understood errors (van Leeuwen 2005;van Leeuwen &Fantino 2005).

The discrepant Hipparcos result for the Pleiades subse-quently led to many efforts to determine the cluster’s distance from binaries and independent parallax measurements (e.g.,Munari et al.2004;Pan et al.2004;Johns-Krull &Ander-son 2005;Soderblom et al.2005).These results support the longer distance scale from MS ?tting,verifying that the Hip-

2An et al.

parcos result was in error.With a formal error of～1%from these measurements,the Pleiades represents a second system (besides the Hyades)with a suf?ciently accurate distance for a precision test of stellar evolutionary models.

Even though the controversy over the Pleiades distance is now settled,a critical assessment of the MS?tting technique is still required to reliably estimate a distance.In fact,MS?t-ting using theoretical isochrones is a complex process that in-volves both physical and empirical considerations(e.g.,Stauf-fer2001).There are,however,many opportunities to check the construction of the isochrones.Stellar evolution models can be tested against the Sun and other stars,such as eclipsing binaries,that have accurate masses and radii.Furthermore, multicolor photometry in nearby clusters and?eld stars can be used to test the bolometric corrections and color-effective temperature(T eff)relations to transform theoretical quantities (luminosity and T eff)to magnitudes and colors(e.g.,Vanden-Berg&Clem2003).

In our?rst two papers of this series(Pinsonneault et al. 2003,2004,hereafter Paper I and Paper II,respectively),we began a long-term effort to assess the accuracy of distances from MS?tting and to reduce or eliminate systematic er-rors in the process,particularly those arising from the trans-formation of theoretical to observational quantities.In Pa-per I,we demonstrated that stellar models from the Yale Ro-tating Evolutionary Code(YREC;Sills et al.2000)are in good agreement with masses and luminosities for the well-studied Hyades eclipsing binary vB22(Torres&Ribas2002). These models also satisfy stringent tests from helioseismol-ogy,and predict solar neutrino?uxes in line with observa-tions(Basu et al.2000;Bahcall et al.2001;Bahcall&Pin-sonneault2004).In Paper II,we showed that the models pro-vide a good match to the spectroscopically determined tem-peratures(Paulson et al.2003)for individual Hyades mem-bers with good parallaxes(de Bruijne et al.2001).However, we found that any of the widely-used color-T eff relations(e.g., Alonso et al.1995,1996;Lejeune et al.1997,1998)fail to reproduce the observed shapes of the MS in the Hyades;dif-ferences in broadband colors were as large as～0.1mag.The existence of these systematic errors in the colors in the pres-ence of agreement between the spectroscopic and theoretical L?T eff scales strongly implies that there are problems with the adopted color-T eff relations instead of errors in the theoretical T eff scale.Therefore,we proposed empirical corrections to the color-T eff relations from Lejeune et al.(1997,1998)that were adopted in the isochrone computations.

In this study,we generate a set of isochrones over a wide range of age and metallicity,and test the validity of the Hyades-based color-T eff corrections using extensive multi-color photometry of four well-studied nearby open clusters. We show that isochrones employing the Hyades empirical corrections precisely match the observed MS shapes,except where anomalously blue colors in young open clusters have been previously noted(Stauffer et al.2003).Furthermore,we demonstrate that the empirical corrections improve the inter-nal precision of the isochrones by examining the consistency of distances derived from several color-magnitude diagrams (CMDs).

We also assess various sources of systematic errors in the MS-?tting technique.Previously,Pinsonneault et al.(1998) considered the effects of age,metal abundance,helium,red-dening,and systematic errors in the photometry,demonstrat-ing that these could not explain the short distance to the Pleiades from Hipparcos.Terndrup et al.(2002)paid atten-tion to the adopted reddening law in a discussion of the dis-tance to NGC2516.Here we extend the error analysis more quantitatively,emphasizing photometric calibration issues and the bias in distance estimates induced by the presence of unresolved cluster binaries or?eld foreground/background stars.

This paper also explores the effect of metallicity on the luminosity of the MS.Metallicity changes isochrone lumi-nosities more strongly than many other input parameters,and the degree of sensitivity depends on the color index used. This permits a purely photometric derivation of the metal-licity(e.g.,Pinsonneault et al.1998,2000;Stello&Nissen 2001;Terndrup et al.2002),which can be compared to metal-licities derived from high-resolution spectra.An agreement between the photometric and spectroscopic metallicities,as we?nd in this paper,provides supporting evidence that the effects of metallicity on the theoretical quantities(L,T eff)and on the color-T eff relations are correctly computed.

The distances in this paper are tied to the Hyades distance at(m?M)0=3.33±0.01(d=46.34±0.27pc),the cluster’s center-of-mass inferred from the Hipparcos catalog(Perry-man et al.1998).Unlike the controversial Hipparcos distance to the Pleiades,the large angular diameter of the Hyades on the sky makes the cluster parallax less vulnerable to the spatial correlation of the Hipparcos parallax(Narayanan&Gould 1999a,b;de Bruijne et al.2001).

In§2we compile cluster photometry,metallicities,red-dening estimates,and information on binarity and member-ship,and present a metallicity for Praesepe from new high-resolution spectra.In§3we brie?y describe the construction of the isochrones.In§4we compute the distances to the sample clusters with the reddening?xed at previously known values and demonstrate that the empirical corrections improve the internal precision of the isochrones.In§5we simultane-ously solve for the cluster metallicity,reddening,and distance from theχ2minimization.In§6we evaluate the effects of several systematic error sources,including those from cluster binaries and?eld star contamination.In§7we discuss sev-eral implications of our results.In the Appendix we address issues on the photometric zero points of the empirical Hyades isochrone.

2.OPEN CLUSTER DATA

2.1.Selection of Clusters

We consider four nearby Galactic open clusters with exten-sive multicolor photometry:Praesepe(=M44;NGC2632), the Pleiades(=M45),M67(=NGC2682),and NGC2516. The choice of these clusters was motivated by several factors. All have well-determined estimates of metal abundance and reddening against which we can compare photometrically-derived values.Samples in Praesepe,the Pleiades,and M67 are dominated by known cluster members,so we can exam-ine whether the Hyades-based color calibration from Paper II generates isochrones that precisely match the shapes of the MS in these clusters.Praesepe has extensive information on binarity,so systematic errors in distances arising from the presence of binaries can be explored.M67and NGC2516 each have modern photometry from two independent studies, from which we gauge the sizes of errors that arise from pho-tometric calibration issues.In addition,NGC2516has a rel-atively high reddening compared to the other clusters,which allows us to evaluate the consequences of adopting particular reddening laws.

The Pleiades is a special case because its distance has re-

Distances from Main Sequence Fitting.III3

TABLE1

R ECENT M EASUREMENTS OF THE P LEIADES D ISTANCE

Reference Method(m?M)0

Narayanan&Gould(1999b)Moving cluster5.58±0.18

Gatewood et al.(2000)Ground-parallax5.58±0.12

Makarov(2002)Hipparcos reanalysis5.55±0.06

Munari et al.(2004)Eclipsing binary(HD23642)5.60±0.03

Pan et al.(2004)Astrometric binary(Atlas)5.65±0.03

Zwahlen et al.(2004)Astrometric binary(Atlas)5.60±0.07

Johns-Krull&Anderson(2005)HST parallax5.66±0.06

Soderblom et al.(2005)HST parallax5.65±0.05

Southworth et al.(2005)Eclipsing binary(HD23642)5.72±0.05

Weighted mean···5.63±0.02

cently been accurately measured from astrometric and eclips-ing binary studies and from ground-and space-based paral-laxes,allowing a precise test of distances derived from MS ?tting.Individual measurements of these studies are summa-rized in Table1,and the weighted average distance from these measurements is(m?M)0=5.632±0.017.The Hipparcos distances to the Pleiades and other clusters are discussed in §7.2.

We use cluster ages from Meynet et al.(1993)for the Pleiades(100Myr),M67(4Gyr),and NGC2516(140Myr). Praesepe is generally considered to be the same age as the Hyades(e.g.,Mermilliod1981b);in Paper II,we assumed an age of550Myr for the Hyades,as derived from isochrones without overshooting(Perryman et al.1998).As we demon-strate,the cluster distances are insensitive to the choice of age.

2.2.Photometry

2.2.1.Praesepe and the Pleiades

We compiled optical photometry for Praesepe and the Pleiades mainly from WEBDA(Mermilliod&Paunzen 2003)5and the Open Cluster Database.6Data in V and B?V for Praesepe came from Johnson(1952),Dickens et al. (1968),and Castelaz et al.(1991).Following the suggestion by Dickens et al.(1968),we added+0.002to V and+0.006 to their B?V to match the Johnson(1952)data.We adopted the photometry from Castelaz et al.(1991)without alteration; the average difference in B?V is+0.002±0.012,Johnson (1952)being redder,among39stars in common(including nonmembers).We compared the numerous sources of V and B?V in the Pleiades,but did not?nd statistically signi?-cant differences in any sample.At B?V 0.8,the scatter in measurements from several sources often exceeded the stated photometric errors;this may result from brightness and color changes from stellar spots on rapidly rotating stars in young open clusters(Stauffer et al.2003).For stars in this color range,as everywhere else,we simply averaged the several available colors and magnitudes.

The situation in V?I C is less straightforward because data for Praesepe and the Pleiades,like those for the Hyades,came from either the Johnson(V?I J)or the Kron(V?I K)sys-tems.The colors on the Johnson system were transformed to the Cousins(V?I C)system using an updated transforma-tion equation as described in the Appendix.The Kron colors were transformed to the Cousins system using the cubic poly-nomial derived by Bessell&Weis(1987).

5See http://obswww.unige.ch/webda/webda.html.

6See https://www.doczj.com/doc/9017065911.html,/staff/stauffer/opencl/index.html.

In Praesepe,stars with V<12have V?I photometry on the Johnson system from Mendoza(1967)and Castelaz et al. (1991),while fainter stars have photometry on the Kron sys-tem from Upgren et al.(1979),Weis(1981),Stauffer(1982b), and Mermilliod et al.(1990).Intercomparisons showed that the Mendoza(1967)and Castelaz et al.(1991)colors were on the same system.The same is true for the Kron data except that the photometry from Upgren et al.(1979)required a red-ward shift of+0.03in V?I K to match the other photometry.In the Pleiades,stars with V<10have V?I photometry on the Johnson system from Johnson et al.(1966),Mendoza(1967), and Iriarte(1969),while fainter stars have photometry on the Kron system from Stauffer(1982a),Stauffer(1982c),Stauf-fer(1984),Stauffer&Hartmann(1987),Stauffer et al.(1989), Prosser et al.(1991),and Stauffer et al.(1994).Direct com-parison between the Mendoza(1967)photometry and that of Iriarte(1969)showed that they agree well for V?I J≤0.5, but the Mendoza(1967)data are systematically redder by 0.02mag for stars with V?I J≥0.5.We opted to adjust the red Mendoza(1967)photometry to place them on the Iriarte (1969)scale.

2.2.2.M67and NGC2516

For M67,we used BV I C photometry from Montgomery et al.(1993,hereafter MMJ93)and also from Sandquist (2004,hereafter S04);these are analyzed separately.S04pro-vided a comparison between the two,revealing statistically signi?cant differences between the two studies.In Table2we compile the mean differences in the photometry for M67and for other clusters as is discussed below.The?rst column of the table lists the cluster name,and the sense of the compar-ison is shown in the second column.The last three columns display the mean difference and its standard error in V,B?V, and V?I C,respectively.In comparison to S04,the MMJ93 data are fainter in V and are redder in V?I C.As shown in Figure5of S04,the differences are largest for those stars with V?I C 1.0.

For NGC2516,we have independent photometry in BV I C from Jeffries et al.(2001,hereafter JTH01)and from Sung et al.(2002,hereafter S02).Neither study compares their pho-tometry with the other.In Figure1we plot the differences in the photometry from the two studies against right ascension, which reveals a position-dependent difference in V(but not in the colors).The sense of the difference is that the photome-try towards the east is fainter in the S02study.According to S02,photometry in this portion of the sky was obtained on a non-photometric night which was then adjusted to match the data in the rest of their survey using stars also observed under good conditions.As their paper lists only the average values,

4An et al.

TABLE 2

D IFFERENCES IN TH

E P HOTOMETRY

Cluster Comparison ?V ?(B ?V ) ?(V ?I )C M67

MMJ93–S04+0.017±0.002+0.009±0.002+0.022±0.001NGC 2516S02–JTH01

+0.016±0.001?0.003±0.001+0.011±0.001

Hyades Ground –Tycho-1a +0.012±0.002?0.009±0.001···Praesepe Ground –Tycho-1a +0.009±0.007+0.004±0.003···Pleiades Ground –Tycho-1a ?0.017±0.003

+0.000±0.002

···r.m.s.

···

0.015

0.006

0.017

a Computed

for stars with V T ≤

9.

F I

G . 1.—Comparison of NGC 2516photometry as a function of right ascension (in degrees).The differences are in the sense of the S02minus the JTH01photometry.

we chose to use the photometry only from regions obtained entirely under photometric conditions.However,there still remain signi?cant differences,as shown in Figure 2,between the JTH01and the S02photometry even when the eastern data in the latter study are https://www.doczj.com/doc/9017065911.html,pared to the JTH01val-ues,the S02data are brighter in V and redder in V ?I C at the top of the MS,but are fainter in V and bluer in B ?V at the faint end.The mean differences are summarized in Table 2.

2.2.

3.Assignment of Random Errors

Most of the collated photometry lacks errors for individual stars,and those errors reported by MMJ93,JTH01,and S02are typically from a small number of repeat observations.Our MS-?tting procedure (§4.1)?rst removes stars that are sta-tistically separated from the MS,and this in turn requires a suitable error for each star.

In Praesepe and the Pleiades,we divided the data into bins in V ,each 2mag wide,and then computed the median

of

F I

G .2.—Comparison of NGC 2516photometry,excluding stars observed under non-photometric conditions by S02.The differences are in the sense of the S02minus the JTH01photometry.

rms in magnitude and colors for the relatively few stars with multiple measurements.We assigned this median value to all data points in each bin as random photometric errors.For NGC 2516,we followed the same binning procedure,then removed a systematic trend in the differences between the JTH01and S02studies by subtracting a linear function in V .We then computed the rms of the differences in V ,B ?V ,and V ?I C in each bin,divided these by

√

Distances from Main Sequence Fitting.III5

TABLE3

D IFFERENCES IN TH

E T YCHO P HOTOMETRY

Cluster Comparison ?B T ?V T

Hyades Tycho-2–Tycho-1?0.009±0.002?0.006±0.002

Praesepe Tycho-2–Tycho-1?0.018±0.004?0.025±0.004

Pleiades Tycho-2–Tycho-1?0.014±0.003?0.016±0.003

N OTE.—Computed for stars with V T≤9.

in MMJ93.We set the errors for the MMJ93study from the rms of the differences with respect to the S04data.

2.2.4.2MASS K s

We calculated V?K s colors from the All Sky Data Release of the Two Micron All Sky Survey(2MASS)Point Source Catalog(PSC).7Here K s designates the“short”-K?lter in 2MASS(Carpenter2001).Based on PSC?ag parameters, we excluded stars that were saturated or undetected.We also ignored blended or contaminated sources.The V?K s errors were taken as the quadrature sum of V errors and the catalog’s “total”photometric uncertainties in K s.

2.3.Systematic Errors in the Photometry

Our analysis of systematic errors in the MS-?tting method (§6.1)shows that calibration errors in the photometry con-tribute signi?cantly to the overall error budget.In addi-tion to the direct comparisons between the studies of M67 and NGC2516,we can use the Tycho-1photometry(van Leeuwen et al.1997)to check whether photometry of the Hyades,Praesepe,and the Pleiades is on a consistent system (the other clusters are too distant to have many stars in the Tycho catalog).Here we assume that the Tycho photometry is consistently on the same scale in all parts of the sky.In Table2,we display the mean differences between the ground-based photometry for these clusters and the Tycho values.The latter were transformed to the Johnson system using the equa-tions in Oja&Evans(1998)for V and in Mamajek et al. (2002)for B?V.The values in the table were computed for stars with V T≤9.

The comparison between the Tycho-1values and the ground-based data is shown in Figure3.The scatter is usu-ally dominated by errors in the ground-based photometry for V≤8and by the Tycho photometry for fainter stars.The solid line in each panel denotes equality,while the dashed line shows the mean difference from Table2.

In Table3we list the differences between the Tycho-1and Tycho-2data(H?g et al.2000)for bright stars in the Hyades, Praesepe,and the Pleiades.The differences were computed directly with the Tycho values(again for V T≤9),without transformations to the Johnson system.Taken together,the data in Tables2and3show that the various studies probably could have calibration errors on the order of0.01–0.02mag in V,but smaller in B?V.The situation in V?I C is less well determined,mainly because we have fewer studies to com-pare.

In Paper II we also noted that the Tycho-1values for V were brighter than the ground-based data by+0.012;we opted to add this constant to the Tycho data before averaging with the ground-based photometry.Had we chosen to adopt the space-based scale instead,our calibrated isochrones would be 7See https://www.doczj.com/doc/9017065911.html,/2mass/.

TABLE4

S PECTROSCOPY IN P RAESEPE

T eff log gξ

ID a(K)(cm s?2)(km s?1)log?σlog?[Fe/H]

KW235700 4.50.57.780.050.10

KW585850 4.50.57.800.060.12

KW3045625 4.40.47.800.060.12

KW3365600 4.50.47.790.060.11

a KW=Klein Wassink(1927).

brighter in V and also bluer in V?K s since the V and K s data were obtained independently.This correction was not applied to the isochrones used in this paper.In the Appendix we dis-cuss other issues involved in the isochrone calibration.

2.4.Membership and Binarity

We also collated information on binarity and membership in Praesepe and the Pleiades from WEBDA and the Open Clus-ter Database.Any star that was listed as a nonmember for any reason(e.g.,from photometry or radial velocities)was rejected.In Praesepe,there are extensive data on binarity (e.g.,Mermilliod&Mayor1999;Bouvier et al.2001).Stars that were designated as likely or probable binary stars were ?agged.In the Pleiades,a considerable number of stars have binarity and membership information(Jones1981;Stauffer et al.1991;Mermilliod et al.1992;Schilbach et al.1995; Bouvier et al.1997;Belikov et al.1998;Moraux et al.2001). We rejected any star with a membership probability less than 50%but kept any star that did not have a membership prob-ability.S04provides a list of single cluster members based on proper motion surveys and known binary systems in the literature and further elimination of unrecognized binaries in CMDs from his high-precision photometry.We used this in-formation for both MMJ93and S04photometry in this paper, unless otherwise speci?ed.

2.5.Metallicity and Reddening

2.5.1.New Spectroscopy in Praesepe Obtaining accurate distances relative to our calibrating clus-ter,the Hyades,requires accurate relative metallicities be-cause the luminosity of the MS is sensitive to the metal abun-dance.As part of this study,we determined a new metallicity for Praesepe using the same method used to?nd the Hyades abundance(Paulson et al.2003).The relative abundance of Praesepe with respect to the Hyades should therefore be accu-rate since systematic errors arising from the solar abundances, employed model atmospheres,oscillator strengths,or effec-tive temperature scales would be minimized.

We obtained spectra of four Praesepe stars with the Magel-lan Inamori Kyocera Echelle(MIKE)spectrograph(Bernstein et al.2003)on the Magellan6.5m Clay telescope at Las Cam-panas Observatory.A0.35′′wide slit gave a resolving power of～55,000per resolution element(4pixels)and wavelength coverage from4500to9200?.The spectra are of high qual-ity,with S/N>100.

The spectra were reduced using standard IRAF8packages. The stellar parameters–T eff,surface gravity(log g),microtur-8IRAF is distributed by the National Optical Astronomy Observatory, which is operated by the Association of Universities for Research in Astron-omy,Inc.,under cooperative agreement with the National Science Founda-tion.

6An et

al.

F I

G . 3.—Differences in V (top )and B ?V (bottom )between the ground-based and the Tycho-1photometry,after transforming the latter into the Johnson system (see text).The differences are in the sense of the ground minus the Tycho-1values.The solid line indicates equality,while the dashed line is plotted at the weighted mean difference for stars with V T ≤9(Table 2).The scatter for V >8mainly re?ects errors in the Tycho photometry.

bulence (ξ),and [Fe/H]–were derived following the proce-dure in Paulson et al.(2003),and are listed in Table 4.We employed the spectral synthesis code MOOG (Sneden 1973)and used stellar model atmospheres based on the 1995ver-sion of the ATLAS9code (Castelli et al.1997).Within IRAF,we used Gaussian ?ts to 15Fe I lines and nine Fe II lines,a subsample of those listed in Table 1of Paulson et al.(2003).The T eff was derived by requiring that individual line abun-dances be independent of excitation potential and that micro-turbulence (ξ)be independent of line strength.Insisting on ionization equilibrium between Fe I and Fe II allowed for the simultaneous determination of log g with T eff and microturbu-lence (ξ).Errors in T eff are ～50K.We analyzed a re?ected light spectrum of the asteroid Iris in order to obtain an instru-mental correction to the solar log ?(Fe).Using this correction,we obtained [Fe /H]=0.11±0.03(s.e.m.9)for the Praesepe cluster.

2.5.2.Metallicity and Reddening Estimates in the Literature

For other clusters in this study,we adopted or calculated av-erage metallicities from high-resolution spectroscopy as sum-marized in Table 5.For the studies by Boesgaard &Friel in Praesepe and the Pleiades,we have recomputed the clus-ter averages from individual [Fe/H]estimates after excluding cluster nonmembers.The only existing measurement of the abundance of NGC 2516using high-resolution spectroscopy is from Terndrup et al.(2002)which was derived from only two stars.

Table 6lists reddening estimates from a variety of sources.For the Pleiades,we adopted a uniform reddening inferred from the H I hole measurement by Breger (1986).Although small parts of the cluster show higher reddening (e.g.,Eggen 1950;Mitchell &Johnson 1957),the stars in these regions constitute only a small fraction of our Pleiades sample,and most of these were tagged as outliers from our ?ltering algo-rithm (§4.1).We compared magnitudes and colors derived

9

Standard error of the mean.

TABLE 5

C LUSTER M ETALLICITY FROM H IGH -R ESOLUTION S PECTROSCOPY

Reference

[Fe/H]s.e.m.Praesepe

Boesgaard &Friel a

+0.090.03This paper (adopted value)

+0.110.03Pleiades

Cayrel et al.(1988)+0.130.07Boesgaard &Friel b +0.030.02King et al.(2000)+0.060.05Weighted mean

+0.040.02···(0.03)c M67

Garcia Lopez et al.(1988)+0.040.04Hobbs &Thorburn (1991)?0.060.03Friel &Boesgaard (1992)+0.020.12Tautvai?ien˙e et al.(2000)?0.030.01Yong et al.(2005)+0.020.02Randich et al.(2006)+0.030.01Weighted mean

+0.000.01···

(0.03)c NGC 2516

Terndrup et al.(2002)+0.01

0.07

a From

Boesgaard (1989)and Friel &Boes-gaard (1992)after excluding known nonmem-bers.

b From Boesgaard (1989)and Boesgaard &Friel (1990)after excluding known nonmem-bers.

c Standar

d deviation of th

e measurements.

from individual E (B ?V )corrections for 157stars (Breger 1986;Stauffer &Hartmann 1987;Soderblom et al.1993)with those from the uniform reddening,but the differences were negligible in our MS ?tting range (§4.2).The reddening mea-surements for the Pleiades from other studies in Table 6likely represent a cluster average including the CO cloud region.

3.CALIBRATED ISOCHRONES

Distances from Main Sequence Fitting.III7

TABLE6

C LUSTER R EDDENING IN THE L ITERATURE

Reference Method E(B?V)s.e.m.

Praesepe

Mermilliod(1981a)UBV0.000.01a

Nicolet(1981)Geneva0.0110.005

Nissen(1988)uvby–βb0.0070.002

Weighted mean···0.0070.002

······(0.003)c

Pleiades

Mermilliod(1981a)UBV0.040.01a

Nicolet(1981)Geneva0.0620.005

Breger(1986)d Sp-type,uvby–βb0.0440.003

Nissen(1988)uvby–βb0.0390.005

Weighted mean···0.0460.005

······(0.009)c

Breger(1986)[adopted value]e Sp-type,uvby–βb0.0320.003

M67

Taylor(1980)Multiple approaches0.0460.006

Eggen(1981)uvby–βb0.0500.013

Janes&Smith(1984)DDO0.0560.006

Burstein et al.(1986)BH map f0.035g0.005

Nissen et al.(1987)uvby–βb0.0320.006

Hobbs&Thorburn(1991)T eff–(B?V)00.0650.018

Montgomery et al.(1993)UBV0.050.01

Schlegel et al.(1998)Dust map0.0320.005h

Weighted mean···0.0410.004

······(0.010)c

NGC2516

Eggen(1972)UBV0.1250.025

Feinstein et al.(1973)Sp-type,UBV0.1160.004

Snowden(1975)uvby–βb0.1200.014

Mermilliod(1981a)UBV0.110.01a

Nicolet(1981)Geneva0.1140.012

Eggen(1983)uvby–βb0.1180.004

Verschoor&van Genderen(1983)V BLUW0.1270.005

Nissen(1988)uvby–βb0.1090.007

Dachs&Kabus(1989)Sp-type,UBV0.120.004

Sung et al.(2002)UBV0.1120.006

Weighted mean···0.1170.002

······(0.005)c

a Assigned value in this paper.

b E(B?V)=E(b?y)/0.74(Cousins&Caldwell1985).

c Standar

d deviation of th

e measurements.

d Averag

e over the cluster?eld.

e Average over the HI hole(see text).

f Burstein&Heiles(1982).

g Corrected for the systematic difference with Schlegel et al.(1998)of0.02mag.

h Quoted precision(16%).

We used the YREC to construct stellar evolutionary tracks at?0.3≤[Fe/H]≤+0.2in0.1dex increments.At each metal abundance,we ran a grid of masses from0.2to8M⊙in 0.05?1M⊙increments depending on the stellar mass.The scaled solar abundance mix from Grevesse&Sauval(1998) and the helium enrichment parameter?Y/?Z=1.2from non-diffusion models were used;see Sills et al.(2000)and Paper I for detailed information.The tracks were interpolated to generate theoretical isochrones at stellar ages from20Myr to4Gyr.

Stellar luminosities and T eff were initially converted to V, B?V,V?I C,and V?K s from the relations by Lejeune et al. (1997,1998).To obtain?ner grids of the isochrones,we in-creased the M V spacing of the isochrone points using a cu-bic spline to match the?M V=0.05spacing of our tabulated color-T eff corrections in Paper II.We then linearly interpolated the isochrones in both[Fe/H]and age at a?xed M V.We?-nally applied the empirical color corrections,as de?ned in Pa-per II and updated in the Appendix of this paper.We assumed that the correction table is valid over the entire model grid in [Fe/H]and age.

The isochrones constructed in this way are available online. 10

4.DISTANCES AT FIXED REDDENING

We now proceed to derive the cluster distances,adopting the reddening values from Table6.We do this in two ways, either holding the metallicity?xed at the spectroscopic values (Table5)or determining a photometric metallicity that brings the distances from different CMDs into agreement.In either case,we show that the calibrated isochrones improve the in-ternal precision of distance estimates from the three CMDs with B?V,V?I C,and V?K s as color indices,and V as a lumi-nosity index;hereafter(B?V,V),(V?I C,V),and(V?K s,V), respectively.

10See https://www.doczj.com/doc/9017065911.html,/iso/.

8An et al.

4.1.Photometric Filtering

Before?tting isochrones,we identi?ed and removed stars

that are far from the MS in comparison to the sizes of

their photometric errors.Such stars could be either fore-ground/background objects or cluster binaries that stand off

from the MS.

The?ltering process iteratively identi?es the MS as the lo-

cus of points of maximum density on the CMDs,indepen-dently of the isochrones.As an initial step,we identi?ed

the MS by hand,and then removed stars more than1mag

away in V.This was necessary in particular for the analysis

in NGC2516,which shows many faint background stars in

its CMDs(see below)that would complicate?nding the den-

sity maxima.The generous rejection criterion of1mag was chosen so that all cluster binaries would be preserved at this stage.

At each step of the iteration,data points on each CMD were sorted by V magnitude into non-overlapping groups:each

group contained

√

σ2V,i+(γiσc,i)2+σ20

,(1)

where?V i is the V difference between the i th data point and MS at the color of the star andσV,i andσc,i are the photomet-ric errors in V and color,respectively.The error in the color contributes to the error in?V i by the slope of the curve,γi, which was evaluated at the star’s color.Because the above constructed curve would not precisely trace the observed MS in the presence of outliers,we addedσ0in quadrature to the propagated photometric errors in the denominator.We ad-justed the value ofσ0so that the totalχ2is equal to N.Ini-tially,we rejected all data points as the MS outliers if theχ2i (the individual contribution toχ2)is greater than9(corre-sponding to a3σoutlier).We repeated adjustingσ0and re-jecting outliers with the reduced set of data points until there remained no point withχ2i greater than the above threshold value.We combined the results from all three CMDs and re-jected stars if they were tagged as an outlier from any one of the CMDs.

We imposed an additional condition for the conver-gence of the algorithm because cluster binaries and fore-ground/background stars are often substantial populations near the MS.We compared the rms of?V i with its median of all absolute deviations(MAD).The MAD of a quantity x i is de?ned by

MAD=1.483median(|x i?median(x i)|),(2) where the correction factor1.483makes the estimator con-sistent with the standard deviation for a normal distribution (Rousseeuw1990).The MAD is a more robust estimator of the dispersion than the standard deviation in the presence of outliers.Therefore,we reduced the size of theχ2i thresh-old value if the fractional difference between the rms and the MAD of?V i is larger than5%,assuming median(?V i)=0. We repeated the above?ltering steps by constructing a new MS curve from the remaining data

set.

F IG. 4.—Result of the?ltering process in Praesepe.All known cluster binaries were included to test the?ltering algorithm.The plus signs are re-jected points from the?ltering,and open circles are those remaining.Filled circles are known binaries that remained after the?ltering.Stars brighter than V=7were excluded before the?ltering algorithm was applied.

Figure4illustrates the result of the?ltering algorithm in Praesepe.Here we include all known cluster binaries but ex-clude stars with a low membership probability(§2.4).We used all three CMDs in the?ltering,but only the(B?V,V) diagram is shown.The plus signs are rejected points from the?ltering,and open circles are those remaining.Some of the stars on the MS were rejected because they were?ltered in other CMDs.Although many binaries remain after the?l-tering(?lled circles),their proximity to the MS would have a minor impact on the derived distance.The bias in the dis-tance due to the remaining binaries is discussed in§6.2from arti?cial cluster CMD tests.

4.2.Isochrone Fitting and the Photometric Metallicity The cluster distances were found by?tting isochrones over a range of metallicity with the adopted age of each cluster (§2.1).The isochrones were reddened to the E(B?V)values in Table6.We adopted a reddening prescription for the broad-band colors by Bessell et al.(1998),who have assumed the ex-tinction law of Mathis(1990).Their reddening formulae were computed from ATLAS9synthetic stellar photometry for a wide range of T eff,and include color-dependent reddening re-sulting from shifts in the effective wavelengths of broadband ?lters.The Bessell et al.formulae give reddening and ex-tinction values for E(B?V)=0.30,so we linearly rescaled them according to the assumed cluster reddening.The color

Distances from Main Sequence Fitting.III

9

F I

G .5.—Example of determining a photometric metallicity in the Hyades.Open circles are the distances derived from CMDs with B ?V ,V ?I C ,and V ?K s as a function of isochrone metallicity.The solid lines connecting these points are least-squares ?ts,and the labels to the left of the lines indicate the corresponding color index.An age of 550Myr and E (B ?V )=0.000were assumed.The photometric metallicity is de?ned as the average [M/H]where the B ?V line crosses the V ?I C and V ?K s lines.

transformation by Carpenter (2001)11was used to compute the reddening in V ?K s .For zero-color stars,we found R V ≡A V /E (B ?V )=3.26,R VI ≡E (V ?I C )/E (B ?V )=1.32,and R V K ≡E (V ?K s )/E (B ?V )=2.91.For stars in the middle of our MS-?tting range,(B ?V )0=0.8,we found R V =3.44,R V I =1.37,and R V K =3.04.

For a given isochrone,we computed each individual star’s distance modulus (μi ),and de?ned the cluster distance mod-ulus in each CMD as the “unweighted”median of μi ,i.e.,(m ?M )0≡median(μi ).We computed the ?tting error in the distance modulus on each CMD as

σ(m ?M )=max

σphot ,

MAD(μi )

N ,(3)where N is the total number of data points used in the ?t.Here

1

σ2V ,i

+(γi σc ,i )2,(4)

where the same notation is used as in equation (1)except that

γi is the isochrone slope at the color of the star.We further computed a “weighted”mean and a “weighted”median of the distance modulus,and discuss the difference between the three distance estimates along with other systematic errors in §6.1.

For a given set of [M/H],age,and E (B ?V ),we ?t isochrones in (B ?V ,V ),(V ?I C ,V ),and (V ?K s ,V )over 0.4≤(B ?V )0≤1.3.This color range is where the Hyades calibration is most reliable (Paper II).There were only a few Hyades members blueward of the range,and the cool end was set by the magnitude limit of the Hipparcos mis-sion.The range corresponds to 0.48≤(V ?I C )0≤1.48and 0.98≤(V ?K s )0≤3.16.For non-zero reddening,the ?tting ranges were made correspondingly redder.

We de?ne the photometric metallicity [M /H]E as the one that brings distances from three CMDs into statistical agree-ment (e.g.,Pinsonneault et al.1998;Stello &Nissen 2001;

11

Updated color transformations 2MASS All-Sky Data Release can be found at

https://www.doczj.com/doc/9017065911.html,/2mass/releases/allsky/doc/sec6_4b.html.

Terndrup et al.2002).Figure 5shows how this process works for the Hyades,our calibrating cluster.The open circles dis-play the derived distances as a function of isochrone metal-licity in each of the three CMDs with E (B ?V )=0.000at an age of 550Myr.The lines connecting these points are least-squares ?ts,and the labels to the left of the lines indi-cate the corresponding color index.The slope in the (B ?V ,V )is larger than in the other two CMDs,indicating a greater sensitivity of the isochrone luminosity to the metal abun-dance.We de?ne [M /H]E as the weighted average of the two metallicities at which the (B ?V ,V )distance agrees with that from (V ?I C ,V )and from (V ?K s ,V ),respectively.For the Hyades in Figure 5,we derive the photometric metallicity of [M /H]E =+0.13±0.02,which is naturally the same as the originally assumed value in the isochrone calibration.

With this de?nition of the photometric metallicity,we can derive two distances at the adopted reddening.The ?rst of these,designated as (m ?M )0,S ,is the weighted average dis-tance modulus from the three CMDs at the spectroscopic [Fe/H].The second distance,(m ?M )0,E ,is the value deter-mined at the photometric metallicity,[M /H]E .

4.3.Results

Figures 6-9display the CMDs for each cluster.For Prae-sepe and the Pleiades,known binaries and nonmembers were excluded before we applied the photometric ?ltering.Stars that remained after the ?ltering are shown as open points,and the rejected stars are shown as plus signs.With the exten-sive binary and membership information,the CMDs of Prae-sepe (Fig.6)are dominated by single cluster members,and there are few stars rejected by the photometric ?ltering.The CMDs of the Pleiades (Fig.7)show that the ?ltering routine identi?ed likely cluster binaries effectively.The single cluster members from S04are shown in the CMDs of M67(Fig.8),none of which were rejected by the ?ltering algorithm.The CMDs of NGC 2516(Fig.9)are from JTH01and have a large number of foreground/background stars as expected in an area survey (for clarity,the point size for the rejected stars was re-duced on the CMDs as compared to the other plots).

Isochrones at the spectroscopic metallicities are overlaid as solid lines in each ?gure,and in most cases they are in excel-lent agreement with the observed shapes of the MS.In partic-ular,the match to the Praesepe MS (Fig.6),which has an age and metallicity nearly identical to the Hyades,is good over almost 7mag in V .We interpret this agreement as indicating that the Hyades-based corrections to the color-T eff relations de?ned in Paper II apply to all clusters,or at least to those with metallicities not too different from that of the Hyades.There are a few cases in which the match to the MS shape is not as good as it is in Praesepe.In the Pleiades (Fig.7),stars with B ?V 0.9are considerably bluer than the isochrone in (B ?V ,V ),although not in the other two CMDs.This “blue K dwarf”phenomenon was discussed by Jones (1972),Landolt (1979),and van Leeuwen et al.(1987),and at greater length by Stauffer et al.(2003),who attributed it to stellar temper-ature inhomogeneities caused by cool spots and plage areas in rapidly rotating young stars.From the JTH01photometry,it would appear that NGC 2516,which is about 40%older than the Pleiades,does not share this phenomenon with the Pleiades (see Fig.9).However,we show in §7.1that the usage of the S02data comes to a different conclusion with supporting evidence from stellar rotational velocities.The data in (V ?I C ,V )for all clusters are somewhat bluer than the isochrone redward of V ?I C ≈1.6;this indicates a pos-

10An et al.

F IG.6.—CMDs of Praesepe,after excluding known binaries and stars with low membership probability.Plus signs are photometrically rejected data points from the?ltering algorithm,and open circles are those remaining.The solid lines are empirically calibrated isochrones with spectroscopic metallicity(Table5), which were adjusted for the reddening with the literature value(Table6).Fitting ranges are shown as horizontal bars.The arrow denotes the direction of reddening vectors.

F IG.7.—Same as Fig.6,but for the Pleiades.As discussed by Stauffer et al.(2003),the Pleiades K dwarfs are bluer than the given isochrone in(B?V,V),but not in the other two CMDs(see text).The distance modulus in(B?V,V)was derived at0.4≤(B?V)0≤0.8as shown by the shorter horizontal bar.

Distances from Main Sequence Fitting.III11

F IG.8.—Same as Fig.6,but for M67with photometry from S04.Shown are those stars designated as single cluster members by S04.The?t was performed over0.7≤(B?V)0≤1.3and corresponding color ranges in the other two CMDs to avoid the steeply rising part of the upper MS.

F IG.9.—Same as Fig.6,but for NGC2516with photometry from JTH01.The?tting range in(B?V,V)was truncated in the red part as in the Pleiades CMD (see text).

12An et

al.

F I

G .10.—Distance vs.isochrone metallicity for Praesepe with the Leje-une et al.(1997,1998)color-T eff relation (top )and with our new empirical correction (bottom ).The reddening value in the literature (Table 6)was as-sumed.Circles and lines have the same meaning as in Fig.5.Vertical lines show the spectroscopic metallicity and its 1σerror (Table 5).

sible systematic error in the Hyades calibration for these red stars,probably resulting from a paucity of Hyades stars with accurately measured parallaxes in this color range.In M67(Fig.8),there may be a larger deviation because the color cal-ibration employed by S04did not extend as far to the red as this.

The distance was independently derived from each CMD over the color range shown as a horizontal bar in each panel.The blue end of the color ranges for M67was increased to (B ?V )0=0.7to avoid the steeply rising parts of the MS,as the resulting distance would be sensitive to the choice of cluster age.For the Pleiades and NGC 2516,the red end of the color range in (B ?V ,V )was decreased to (B ?V )0=0.8to avoid systematic errors from the blue K dwarf phenomenon.

In Table 7we assemble our derived cluster distances and the photometric metallicities.The ?rst column lists the source of photometry,including particular selections of subsamples of the data.We applied the photometric ?ltering to each sub-sample before ?tting isochrones.The second through fourth columns display the distance modulus found from each CMD,where the spectroscopic metallicity and reddening were taken from Tables 5and 6,respectively.The weighted average dis-tance,(m ?M )0,S ,is shown in the ?fth column.The last two columns display the photometric metallicity,[M /H]E ,and the distance modulus,(m ?M )0,E ,at this metallicity.The error in (m ?M )0,S is either the propagated one from the errors of the individual distances or the s.e.m.of these distance estimates,whichever is larger.The error in (m ?M )0,E additionally in-cludes the propagated error from [M /H]E

.

F I

G .11.—Same as Fig.10,but for M67with MMJ93(top )and S04(bottom )photometry.The stars designated as single cluster members by S04were used in both cases.

The standard errors of the average distance modulus at the spectroscopic metallicities are typically on the order of 0.02mag (i.e.,the individual distances in the three CMDs are con-sistent to ～2%).In Figure 10we show that this consistency is primarily the result of the empirical color-T eff corrections derived in Paper II.The top panel shows the derived distances to Praesepe as a function of the metallicity for isochrones that do not incorporate these corrections,while the bottom panel shows what happens when the corrected isochrones are employed.The vertical lines show the metallicity estimate for Praesepe with its 1σerror (§2.5.1).The internal preci-sion of the distance estimation with isochrones employing the Hyades empirical calibration is about a factor of 5better than with the uncorrected isochrones.

The distances and photometric metallicities in Table 7are generally not sensitive to the selection of subsamples in each cluster.For example in Praesepe and the Pleiades,leaving known binaries in the sample before ?ltering changed the value of (m ?M )0,S by only about 0.01mag,con?rming that the ?ltering algorithm removed most of the binaries that are brighter than the MS (e.g.,Fig.4).In M67,we compared the distances from the MMJ93data,but also using their photom-etry for only those stars identi?ed as single members by S04;we found a negligible difference.In NGC 2516,we computed the distances from the full JTH01or S02catalogs,or select-ing only those stars identi?ed as radial velocity (RV)members (Terndrup et al.2002)or X-ray detected sources (Damiani et al.2003);again making that selection reduced the distance modulus by only about 0.01mag.

Figures 11–13display the derived distances as a function of metallicity for M67,NGC 2516,and the Pleiades,respec-

Distances from Main Sequence Fitting.III13

TABLE7

MS F ITTING D ISTANCE AND P HOTOMETRIC M ETALLICITY AT L ITERATURE E(B?V)

(m?M)0

14

An et al.

for Praesepe at the spectroscopic metallicity also agrees with existing geometric distance determinations:Gatewood &de Jonge (1994)found (m ?M )0=6.42±0.33from ground-based parallaxes,and Loktin (2000)obtained 6.16±0.19from the moving cluster method.In addition,there are two determinations of the distance to Praesepe from Hipparcos :

(m ?M )0=6.37±0.15(van Leeuwen 1999)and 6.28+0.13

?0.12(Ro-bichon et al.1999).We compare our derived distances with the Hipparcos measurements in §7.2.

5.SIMULTANEOUS DETERMINATION OF CLUSTER PARAMETERS

In the previous section,we showed that the cluster metal-licities can be obtained from the MS ?tting because the vari-ous color indices have different sensitivities to the metallicity.However,we can further constrain the reddening because the slope of the MS becomes shallower below M V ≈5.5,espe-cially in (V ?I C ,V )or (V ?K s ,V ).Furthermore,the reddening vectors are not parallel to the MS as shown in Figure 6,so the derived distances depend on the reddening with different degrees of sensitivity in the three CMDs.

To determine the metallicity,reddening,and distance simul-taneously,we minimized

χ2tot

=

3 j =1N j i =1

(μi j ?ˉμ)2

σphot ,j

,1

.

(6)

This renormalization was required in particular for the S04

data,which had many more repeat measurements in V and I C than in B ,so the quoted errors are much smaller in V ?I C than in the other colors.For these data,we found f ≈6in (V ?I C ,V ),indicating that the errors are greatly underesti-mated at least when compared to the scatter about the best-?tting isochrone.Without the renormalization factor,the ma-jority of the weight in the χ2tot would be given to the (V ?I C ,V )CMD,and the process would almost entirely ignore the infor-mation on the other CMDs.

We searched for the minimum χ2tot in the plane of [M/H]versus E (B ?V )using a downhill simplex method (Press et al.1992).The average distance modulus in equation (5)was determined at each [M/H]and E (B ?V ).Because the color ranges used in the ?tting depend on the adopted E (B ?V ),we solved for the minimum χ2tot at the E (B ?V )derived from the previous iteration to keep the same number of data points for each iteration.Since we de?ned χ2tot with respect to the av-erage distance modulus from the three CMDs,the minimum χ2tot yields a set of [M/H]and E (B ?V )that best describes all of the CMDs simultaneously.

Figure 14shows likelihood contours in the [M/H]versus E (B ?V )plane for all clusters in this paper and for the Hyades.The χ2surfaces were smoothed with a Gaussian kernel.Con-tours are shown at ?χ2tot =3.53,8.02,and 14.2relative to the

minimum value of χ2

tot ,which correspond to 68.3%,95.4%,and 99.7%con?dence levels (1,2,3σ)for the 3degrees of freedom,and the errors in both quantities were assumed to be normally distributed without correlation.The vertical lines on each panel display the spectroscopic metallicity and its 1σerror,while the horizontal lines show the average reddening in the literature with its 1σerror.The error bars in each panel show the size of systematic errors,which is discussed in the next section.

The solution for Praesepe (Fig.14,middle left panel )yields a value for the reddening that is slightly negative and a metal-licity that is ～2σhigher than found spectroscopically.The negative reddening may be a consequence of zero-point er-rors in the photometry or else from an error in the Hyades reddening,which we assumed in Paper II as identically zero.The study of Taylor (1980),for example,indicates E (B ?V )=+0.003±0.002for the Hyades.So,if the Praesepe redden-ing was slightly less than the non-zero Hyades reddening,the MS-?tting method would yield a negative value.

In Figure 14two solutions are shown for M67(top mid-dle panel ):the contours with the higher E (B ?V )are from the MMJ93data,and the others are from the S04.The large difference in E (B ?V )obtained from these two samples is pri-marily due to the difference in the V ?I C values (Table 2).We illustrate this further by showing the χ2contours that result when only the B ?V and V ?K s data are included.The mid-dle panel shows the likelihood contours for the MMJ93data,while the bottom middle panel is for the S04.Here the two studies yield the same values of metallicity and reddening,al-though with larger errors.We also display two solutions for NGC 2516in the bottom left panel of Figure 14.The con-tours with the higher reddening are from the SBLL03,and the others are from JTH01photometry.

The top right panel of Figure 14shows the solution for the Pleiades,while the panel below that shows the solution for the metallicity and reddening that result if the average geo-metric distance from Table 1is taken as a prior.Contours in this panel are 1,2,and 3σcon?dence levels for the 2degrees of freedom.This demonstrates that good photometry and par-allaxes together can provide strong constraints on the metal abundance of a cluster.

Table 8lists the results from the χ2solutions for vari-ous combinations of the data listed in the ?rst column.The [M /H]χ2and E (B ?V )χ2denote our derived values of metal-licity and reddening,respectively,and are shown in the third and fourth columns.Errors in these quantities are the sizes of the 1σcontours in Figure 14.The (m ?M )0,χ2in the second column is the average distance from the three CMDs.The errors in metallicity and reddening were propagated into the error in distance.We compare these values with previous es-timates in the literature after a discussion of systematic errors in the next section.

6.SYSTEMATIC ERRORS AND THE ACCURACY OF MS FITTING

In the previous two sections we showed that MS ?tting can be performed with high internal precision,resulting in errors in distance moduli of 0.02mag (i.e.,1%in distance)at the spectroscopic metallicity (Table 7).When spectroscopy is not available,photometry in BV I C K s alone can be used to derive distances with precision of 1%–3%,metallicities with 0.02–

Distances from Main Sequence Fitting.III

15

F I

G .14.—Likelihood contours in [M/H]vs.E (B ?V )shown at ?χ2tot =3.53,8.02,and 14.2(1,2,and 3σfor 3degrees of freedom)relative to the minimum value of χ2tot

(eq.[5]).Those for 2degrees of freedom are shown for the Pleiades at the geometric distance (middle right ).For NGC 2516(bottom left ),contours with the higher E (B ?V )are from the S02,and the others are from the JTH01photometry.For M67(top middle ),contours with the higher E (B ?V )are from the MMJ93,and the others are from the S04photometry.Also shown are contours for the MMJ93(middle )and the S04(bottom middle )in the case where only (B ?V ,V )and (V ?K s ,V )are used.Error bars represent 1σsystematic errors listed in Tables 9-12excluding ?tting errors.The spectroscopic metallicities and reddening values in the literature with 1σerrors (Tables 5,6)are shown with vertical and horizontal lines,respectively.

0.04dex error,and reddening estimates with better than 0.01mag error (Table 8).However,various systematic errors in the MS ?tting should be considered in the ?nal error budget such as the photometric zero-point errors or biases introduced by the cluster binaries and foreground/background stars.In this section we estimate these systematic errors and show that they often exceed the internal precision of the MS ?tting by factors of 2–3.

6.1.Errors from Input Quantities

In Tables 9–12we list the sources of several systematic er-rors and their contributions to errors in the distance,photo-metric metallicity,and reddening estimates for each cluster.The ?rst column displays the source of errors,and the second column shows the size of the errors adopted for each cluster.The third column lists error contributions to (m ?M )0,S when both metallicity and reddening are held ?xed.The fourth and ?fth columns contain errors in (m ?M )0,E and [M /H]E when the photometric metallicity is determined at a ?xed reddening.The sixth through eighth columns list errors in (m ?M )0,χ2,

[M /H]χ2,and E (B ?V )χ2from the χ2minimization.At the bottom of the table,we list the “total error,”computed as the quadrature sum of all of the error contributions.These are the errors that are shown as error bars along with the χ2contours in Figure 14.The ?tting errors (i.e.,the internal precision of MS ?tting)are not included in these sums but instead are sep-arately tabulated below.

In the ?rst column of these tables,the line marked [Fe /H]denotes the error in the adopted cluster metallicity (Table 5).This affects the error in (m ?M )0,S only.The errors in the adopted reddening (Table 6)are shown in the line marked E (B ?V ).These errors are propagated into the determina-tion of the photometric metallicity [M /H]E and the distance (m ?M )0,E ,but not the values determined from the χ2mini-mization.The remaining errors will contribute to the system-atic errors of all of the parameters found via MS ?tting.

We chose an error of 30%in the adopted age of each cluster from the consideration of previous age estimates for the Pleiades.The cluster age is about 100Myr from the

16An et al.

TABLE8

χ2S OLUTIONS

Data/Constraint(m?M)0,χ2[M/H]χ2E(B?V)χ2

Praesepe

known binaries excluded6.365±0.025+0.202±0.020?0.003±0.005

known binaries included6.357±0.026+0.192±0.0200.002±0.004

Pleiades

known binaries excluded5.693±0.053+0.079±0.0400.032±0.007

At geometric distance a5.632±0.017b+0.058±0.0120.024±0.004

known binaries included5.667±0.054+0.059±0.0400.027±0.007

At geometric distance c5.632±0.017b+0.049±0.0120.025±0.004

M67

S049.558±0.034?0.043±0.0320.024±0.006

MMJ939.603±0.051?0.061±0.0350.078±0.006

MMJ93/S04stars d9.638±0.053+0.019±0.0500.061±0.010

NGC2516

JTH018.001±0.038?0.076±0.0200.112±0.007

JTH01/RV+X-ray e7.939±0.059?0.133±0.0400.120±0.010

S028.064±0.038?0.055±0.0300.139±0.007

S02/RV+X-ray e8.038±0.091?0.050±0.0600.134±0.015

N OTE.—All subsamples of data were?ltered independently before?tting

isochrones.

a Solution at the average geometric distance after excluding known binaries.

b Average geometri

c distance from Table1.

c Solution at the average geometric distance with known binaries include

d in the

data.

d Singl

e cluster members from S04with MMJ93photometry.

e RV members from Terndrup et al.(2002),and X-ray detected sources from Dami-

ani et al.(2003).

TABLE9

MS F ITTING E RROR B UDGET FOR P RAESEPE

Adopted[Fe/H]Adopted E(B?V)χ2Minimization Source of Error?Quantity?(m?M)0,S?(m?M)0,E[M/H]E?(m?M)0,χ2[M/H]χ2E(B?V)χ2

[Fe/H].......±0.03±0.029···············

E(B?V)......±0.002±0.005±0.013±0.007·········

Age..........±30%±0.007±0.010±0.000?0.014?0.013?0.004

Helium(Y)...±0.009?0.027?0.027±0.000?0.027±0.000±0.000

Calibration...±0.010±0.020±0.020±0.010±0.010±0.002

Fitting method±0.006±0.014±0.013±0.002±0.004±0.001

R V...........±0.3±0.000±0.006±0.006±0.000?0.001±0.000

R VI..........±0.07±0.001±0.004±0.003?0.002?0.003±0.001

R VK..........±0.12±0.001±0.002±0.001±0.000±0.000±0.000

?V..........±0.006±0.001?0.011?0.011?0.003?0.008±0.002

?K s.........±0.007±0.006±0.019±0.013±0.010±0.010?0.002

?(B?V).....±0.004?0.006±0.038±0.044±0.014±0.027?0.002

?(V?I)C.....±0.006?0.008?0.027?0.021?0.007?0.006±0.002

Total.........±0.044±0.065±0.058±0.037±0.035±0.006

Fitting........±0.021±0.030±0.030±0.025±0.020±0.005

MS turnoff using isochrones that incorporate convective over-shooting(Meynet et al.1993),while ages from the lithium depletion boundary are about125Myr(Stauffer et al.1998; Burke et al.2004).However,distances,photometric metallic-ities,and reddening values are insensitive to age because the color range we chose for the?tting avoids the upper MS and the pre-MS(for clusters at least as old as the Pleiades).

The helium abundance(Y)sensitively affects the isochrone luminosity(?M V/?Y≈3at?xed T eff).A shorter distance is obtained when a higher helium abundance is assumed.We adopted an error in Y of0.009for each cluster.This is a1σscatter in?Y/?Z from the primordial helium abundance,the solar value,and that of the Hyades as inferred from the MS luminosity at the Hipparcos distance(see Paper I).The photo-metric metallicities and reddening values are not affected by the helium abundance since the error in Y equally changes the distances in the three CMDs.

The calibration errors in the tables re?ect errors in our adopted parameters of the Hyades:[Fe/H]=+0.13±0.01 (Paulson et al.2003)and the center-of-mass distance modulus of(m?M)0=3.33±0.01(Perryman et al.1998).We adopted a1σerror in reddening from Taylor(1980),E(B?V)= +0.003±0.002,but assumed a zero reddening toward the Hyades(Paper II).The errors in[M/H]

E

and(m?M)0,E fur-ther include the errors arising from the scatter about the linear relation between the two quantities(see Fig.5).The error from the?tting method comes from experiments in which we used the weighted mean or weighted median distance modu-lus instead of the unweighted median value.

The next three rows in the tables show the effects of er-

Distances from Main Sequence Fitting.III17

TABLE10

MS F ITTING E RROR B UDGET FOR THE P LEIADES

Adopted[Fe/H]Adopted E(B?V)χ2Minimization

Source of Error?Quantity?(m?M)0,S?(m?M)0,E[M/H]E?(m?M)0,χ2[M/H]χ2E(B?V)χ2

[Fe/H].......±0.02±0.023···············

E(B?V)......±0.003±0.007±0.010±0.002·········Age..........±30%+0.006+0.020+0.011+0.004+0.002+0.001

Helium(Y)...±0.009?0.027?0.027±0.000?0.027±0.000±0.000 Calibration...±0.010±0.020±0.020±0.010±0.010±0.002

Fitting method±0.006±0.038±0.033±0.027±0.014±0.005

R V...........±0.3?0.003±0.018±0.018±0.013±0.016?0.001

R VI..........±0.07±0.003±0.012±0.008±0.010±0.008?0.001

R VK..........±0.12±0.002±0.008±0.005±0.007±0.005±0.000

?V..........±0.012±0.008?0.009?0.014?0.013?0.018±0.000

?K s.........±0.007±0.003±0.012±0.008±0.014±0.010±0.000

?(B?V).....±0.008?0.023±0.058±0.069±0.048±0.066?0.002

?(V?I)C.....±0.009?0.013?0.037?0.021?0.034?0.030±0.006 Total.........±0.048±0.092±0.087±0.076±0.080±0.009 Fitting........±0.013±0.039±0.032±0.053±0.040±0.007

TABLE11

MS F ITTING E RROR B UDGET FOR M67

Adopted[Fe/H]Adopted E(B?V)χ2Minimization

Source of Error?Quantity?(m?M)0,S?(m?M)0,E[M/H]E?(m?M)0,χ2[M/H]χ2E(B?V)χ2

[Fe/H].......±0.01±0.008···············

E(B?V)......±0.004±0.008±0.016±0.009·········Age..........±30%±0.006±0.009±0.005±0.001?0.002?0.002

Helium(Y)...±0.009?0.027?0.027±0.000?0.027±0.000±0.000 Calibration...±0.010±0.020±0.020±0.010±0.010±0.002

Fitting method±0.004±0.011±0.016±0.001±0.001±0.000

R V...........±0.3?0.004±0.019±0.026±0.000±0.008?0.002

R VI..........±0.07±0.006±0.014±0.010±0.004±0.002?0.001

R VK..........±0.12±0.001±0.007±0.007±0.002±0.001±0.000

?V..........±0.009±0.006?0.006?0.014?0.007?0.016±0.000

?K s.........±0.007±0.002±0.011±0.011±0.012±0.012±0.000

?(B?V).....±0.005?0.006±0.038±0.049±0.026±0.043?0.003

?(V?I)C.....±0.011?0.022?0.045?0.023?0.003?0.009±0.014 Total.........±0.040±0.076±0.070±0.041±0.050±0.014 Fitting........±0.015±0.054±0.062±0.034±0.032±0.006

N OTE.—Estimated from S04photometry.

TABLE12

MS F ITTING E RROR B UDGET FOR NGC2516

Adopted[Fe/H]Adopted E(B?V)χ2Minimization

Source of Error?Quantity?(m?M)0,S?(m?M)0,E[M/H]E?(m?M)0,χ2[M/H]χ2E(B?V)χ2

[Fe/H].......±0.07±0.072···············

E(B?V)......±0.002±0.006±0.005±0.000·········Age..........±30%±0.005±0.014±0.008±0.014±0.006±0.000

Helium(Y)...±0.009?0.027?0.027±0.000?0.027±0.000±0.000 Calibration...±0.010±0.020±0.020±0.010±0.010±0.002

Fitting method±0.009±0.016±0.019±0.002±0.003±0.000

R V...........±0.3?0.009±0.051±0.058±0.030±0.063?0.010

R VI..........±0.07±0.008±0.017±0.009±0.011±0.005±0.000

R VK..........±0.12±0.009±0.031±0.023±0.016±0.021?0.005

?V..........±0.008±0.002?0.011?0.015?0.004?0.013±0.002

?K s.........±0.007±0.005±0.017±0.013±0.011±0.012?0.002

?(B?V).....±0.002?0.004±0.012±0.016±0.011±0.016±0.000

?(V?I)C.....±0.006?0.007?0.014?0.007?0.009?0.007±0.002 Total.........±0.080±0.079±0.074±0.051±0.072±0.012 Fitting........±0.023±0.058±0.053±0.038±0.020±0.007

N OTE.—Estimated from JTH01photometry.

18An et al.

rors in the reddening laws.Wegner(2003)determined R V towards about600OB stars,from which we estimated an un-certainty of±0.3from the MAD of the distribution.We es-timated the dependence of R V I and R V K on the choice of R V from the extinction law of Cardelli et al.(1989),and found ?R V I/?R V?0.22and?R V K/?R V?0.84for B-type stars. We adopted±0.07for the error in R V I from the difference between our adopted value and R V I=1.24from Dean et al. (1978).Similarly,we adopted±0.12as the error in R V K by considering our value with those advocated by Schultz& Wiemer(1975)and by Sneden et al.(1978).These are signif-icant contributors to the error budget for the highly-reddened cluster(NGC2516),but are much less signi?cant for the other clusters.

The next four rows list the errors resulting from zero point errors in the photometry.In this calculation,we assumed that the fundamental observed quantities are V,K s,B?V, and V?I C(e.g.,Stetson et al.2003).The adopted values of the errors are shown in the second column.The error in K s was taken as the calibration uncertainty(0.007mag)that was speci?ed in the explanatory supplement to the2MASS All Sky Data Release.12For the Pleiades and Praesepe,the errors come from intercomparisons between limited subsets of stars in common among various studies and from a consideration of the entries in Tables2and3.For M67and NGC2516,we assumed the magnitude and color errors to be half the differ-ence between the two studies(Table2).

Besides the systematic errors in the photometry,distance estimation can be affected by the variability of pre-MS and MS stars(e.g.,Scholz&Eisl?ffel2004),which may be caused by rotational modulation of cool and hot regions on stellar surfaces.This issue can be addressed with multiple ob-servations for each star,but those kinds of data are not usually available.

Errors in the metallicity and the helium abundance are im-portant contributors to the error in the distance(m?M)0,S.As shown in Table5,individual spectroscopic studies generally have～0.03dex errors in the mean metal abundance,and this is propagated into the error of～0.03mag in distance modu-lus since?(m?M)0/?[Fe/H]≈1.In M67,the error in the metallicity has no greater effect than the other systematic er-rors due to the many independent metallicity estimates.This contrasts well with the NGC2516case where only two stars were measured from high-resolution spectroscopy.The zero point errors in the photometry dominate the systematic errors in the photometric metallicity and the distance derived purely from the isochrones,independently of the spectroscopy.How-ever,our analysis demonstrates that the photometric metallic-ity and the distance can be derived with comparable accuracy to those from the spectroscopic studies.In all cases,the?tting errors are smaller than the total systematic errors by factors of 2–3as shown in the tables.

The cluster richness can also affect the distance determi-nation.If there are N genuine single cluster members with random photometric errors in colors of?,the error in distance would be approximately s?(3N)?0.5≈2?N?0.5,where s is a typical slope of MS.This error was already included in the?t-ting error,but its contribution is small for clusters with good photometry and N≈100.However,information on single cluster members is only available for a handful of open clus-ters,and the effects of binaries and foreground/background stars should be taken into account in distance estimation,as is 12See https://www.doczj.com/doc/9017065911.html,/2mass/releases/allsky/doc/explsup.html.discussed in the next two sections.

6.2.Cluster Binaries

The?ltering algorithm(§4.1)removes cluster binaries if they are suf?ciently far from the MS.However,low mass ra-tio binaries would remain and might systematically reduce the estimated distance since they are brighter and redder than sin-gle cluster members.

To determine the size of the bias in distance estimation and to evaluate the effectiveness of the?ltering algorithm,we per-formed arti?cial cluster CMD tests using a solar-metallicity isochrone for an age of550Myr.We constructed each set of CMDs by generating200single stars or unresolved binaries in the color range used for the MS?tting,0.4≤(B?V)0≤1.3. Primaries and secondaries in the binaries,as well as single MS stars,were randomly drawn from the mass function(MF)for M35(Barrado y Navascués et al.2001).The lower mass limit for the secondaries was set to be the hydrogen-burning limit at 0.08M⊙.Each simulation is characterized by the binary frac-tion(BF),which is de?ned as the number of binaries divided by the total number of systems in the above?tting range.For example,at BF=50%,one-half of the data points correspond to single stars and the other half represent unresolved binaries. We neglected multiple systems other than binaries due to the observed low frequency among solar-type stars(Duquennoy &Mayor1991).We assigned equal photometric errors of0.01 mag in V,B?V,V?I C,and V?K s and displaced them from the isochrone assuming a normal distribution.The arti?cial CMDs were then?ltered,and the distance was derived in the usual way.

Figure15summarizes the results of these simulations.The top three panels show the bias in the distance modulus as a function of BF.The thick solid line shows the median of the bias from200arti?cial cluster CMDs computed at each BF with intervals of5%.Thin lines on either side are the?rst and third quartiles of these distributions.The dashed line indicates ?(m?M)0=0.The vertical error bar displays the quartile ranges for a BF of50%,but with errors for each star of0.03 mag.

The bottom panel of Figure15displays the ratio of the num-ber of data points remaining after?ltering compared to the in-put number of single stars in the simulation.The lines have the same meaning as in the other panels.For example,at BF =50%,the simulation yields N?t/N single?1.3after?ltering, indicating that～70%of binaries were eliminated by the?l-tering algorithm.

As seen in Figure15,the distance from the(B?V,V)CMD is less affected by the remaining binaries than those from (V?I C,V)and(V?K s,V).Since stars were rejected if they were?ltered at least in one of the CMDs,the total number of remaining binaries is the same in the three CMDs with a bi-nary detection limit set by a certain binary mass ratio13(～0.4 in our simulation).However,binaries stand out more promi-nently in(V?I C,V)or(V?K s,V)than in(B?V,V)because the addition of the cooler secondaries results in a smaller change in the combined color in B?V than in the other two color in-dices.As also seen in Figure15,larger photometric errors result in a bigger bias in distance estimation since binaries are more dif?cult to identify when the photometric errors are larger.

The effects of binaries on the determination of the photo-

13Binary mass ratio is de?ned as the mass of the secondary(less massive) star divided by the mass of the primary star.

Distances from Main Sequence Fitting.III

19

F I

G .15.—Effects of remaining binaries after photometric ?ltering from arti?cial cluster CMD tests (see text for details).Top three panels:Bias in distance modulus as a function of binary fraction in each CMD with assumed photometric errors of 0.01mag.Bottom:Number ratio of all data points used in the ?t to the input single stars.The thick solid line shows the median of these values from 200arti?cial cluster CMDs at each binary fraction with intervals of 5%,and thin lines on either side are the ?rst and third quartiles.The dashed line indicates ?(m ?M )0=0.The differences are in the sense of shorter distances and more remaining binaries at a higher binary fraction.The vertical error bar displays the quartile ranges for a binary fraction of 50%,but with errors for each star of 0.03mag.

metric metallicity and reddening are displayed in Figure 16.The top to bottom panels display the biases in [M /H]E ,[M /H]χ2,and E (B ?V )χ2,respectively.Error bars at BF =50%show the results with the photometric errors of 0.03mag.As the BF increases,the MS ?tting gives lower metallicity estimates.This is a direct result of the systematic errors in distance as seen in Figure 15.Recall that if an isochrone at a particular metallicity yields a distance from (B ?V ,V )that is longer than from (V ?I C ,V )or (V ?K s ,V ),the photometric metallicity is smaller than that of the isochrone.Increasing the BF leaves the distances from (B ?V ,V )relatively unaltered but gives shorter distances in the other two colors,thus yielding the lower photometric metallicity.Nevertheless,the amount of a bias in the photometric metallicity is ?[M /H]χ2≈?0.03at BF =50%.In comparison,we conducted some experiments in which the ?ltering was not performed and found that this reduced the metallicity estimates by ?[M /H]χ2≈?0.30.The results from the bottom panel show that the reddening deter-mination is almost unaffected by the presence of un?ltered binaries.

In Figure 16a small offset of ?[M /H]E ≈0.02at BF =0%shows the systematic error that was already included in the “calibration”error in [M /H]E (Tables 9–12).This

happens

F I

G .16.—Same as Fig.15,but for the bias in [M /H]E ,[M /H]χ2,and E (B ?V )χ2(top to bottom ).The differences are in the sense of a lower metal-licity and a higher reddening estimate at a higher binary fraction.

because we assumed a linear relation between metallicity and distance over a wide range of metallicity (e.g.,Fig.5);higher order terms in the relation may be required to eliminate this systematic offset.

Table 13summarizes the predicted bias in the distance,pho-tometric metallicity,and reddening induced by un?ltered bi-naries at BF =50%,which is typical of open clusters (Pa-tience et al.2002).We compare results by adopting different MFs for secondaries,as listed in the ?rst column.We further tested several combinations of the various MFs including the Kroupa et al.(1993)MF for both primaries and secondaries.However,we found that the detailed form of the MF has no greater impact on the distance determination than the inter-nal precision of photometry.The adopted random errors in the photometry are shown in the second column,while the remaining columns display the systematic errors.

In Praesepe,many solar-type stars have been observed for binarity from IR speckle,direct imaging,and spectroscopic studies,covering a wide range of orbital periods (Mermilliod &Mayor 1999;Bouvier et al.2001;Patience et al.2002).As a result,its MS (Figs.4and 6)is fairly clean,suggesting that our binarity information in this cluster is almost com-plete at least for solar-type stars.In 0.4≤(B ?V )0≤0.8,there are 46known binaries out of 116systems,yielding a BF of 40%.At this value,our simulation results in Figures 15and 16yield ?(m ?M )0,S =?0.006,?[M /H]E =?0.014,?[M /H]χ2=?0.037,and ?E (B ?V )=+0.003.These val-ues are consistent with those from the comparison of two sub-samples,computed before and after excluding known cluster binaries (Tables 7and 8).

The Pleiades has also been extensively surveyed on binarity (e.g.,Mermilliod et al.1992;Bouvier et al.1997).However,there are many stars remaining above the MS even after ex-cluding all of the known binaries,suggesting that many bi-naries were probably left undetected by the previous surveys (see Fig.7).In NGC 2516,there exists limited information on

20An et al.

TABLE 13

B IAS I NDUCED BY U NFILTERED B INARIES FROM A RTIFICIAL

C LUSTER

T EST AT BF=50%MF a σphot ?(m ?M )0?[M /H]E ?[M /H]χ2?E (B ?V )χ2

M35±0.01?0.010?0.007?0.033+0.002M35±0.03?0.028?0.034?0.050+0.003?at

±0.01?0.006+0.004?0.020+0.002Salpeter b

±0.01

?0.010

?0.006

?0.023

+0.000

a Stellar

mass function for secondaries.Primaries were generated from M35

mass function (Barrado y Navascués et al.2001).b Salpeter (1955).

binarity only for B-and A-type stars from the spectroscopic studies (Abt &Levy 1972;González &Lapasset 2000).

To estimate the BF for each cluster,we ?rst counted the to-tal number of stars that were 0.3–1.0mag brighter in V than the cluster MS in (B ?V ,V ).For NGC 2516,we subtracted ?eld foreground/background stars based on the distribution of (m ?M )0(see next section)from the JTH01catalog before counting likely cluster binaries.Within the above ?V range,the observed BFs are 19±5%for the Pleiades and 18±2%for NGC 2516,while that of Praesepe is 19±4%.If we com-bine these estimates with the total observed BF of 40%for Praesepe,the total BFs for the Pleiades and NGC 2516would be 38±11%and 38±4%,respectively.The errors represent those from counting statistics.

Previous studies have also noted that Praesepe and the Pleiades have the similar BF among G-and K-type stars over a certain range of orbital periods (Mermilliod &Mayor 1999;Bouvier et al.2001;Patience et al.2002).However,the above BF estimates based on the observed fraction of stars with ?V ≥0.3are uncertain for several reasons.For example,the distribution of binary mass ratio or the secondary MF can be different from one cluster to another.If we assume a ?at MF for secondaries,we would derive BF ≈50%.On the other hand,the M35MF predicts quite a high BF (>90%)and pro-duces too many binaries near MS (low mass ratio binaries)compared to the observed distribution of stars.It should also be noted that the BF in the simulation only concerns photo-metric binaries.In other words,blending of physically un-related stars could increase the BF,while binaries with large angular separations could reduce the BF.As a result,the BF depends on a number of cluster properties,as well as a spe-ci?c design of a photometric survey.In addition,a substan-tial fraction of binaries tend to have equal-mass companions (e.g.,Halbwachs et al.2003;Pinsonneault &Stanek 2006,and references therein).These binaries would be well-separated from the cluster MS,so they can be easily detected and re-moved by the ?ltering algorithm.The net effect would be an overestimation of the bias in distances and photometric metal-licities in the simulation for a given BF.

From the above considerations,we adopted 30%–50%as the range of ±1σformal errors in the BF for the Pleiades and NGC 2516.From the simulation result (0.01mag error,M35MFs for both binary components),this BF yields ?(m ?M )S =?0.007±0.003,?[M /H]E =?0.020±0.015,?[M /H]χ2=?0.036±0.013,and ?E (B ?V )χ2=+0.003±0.001.These values should be subtracted from the MS-?tting results in Ta-bles 7and 8to correct the effects of binaries.Additional simulations showed that ?[M /H]χ2=+0.008±0.004and ?E (B ?V )χ2=+0.003±0.001in the case of the ?xed clus-ter distance.Bias in distance at the photometric metallicity

would be the quadrature sum of ?(m ?M )S and the error con-tributions from ?[M /H]χ2(or ?[M /H]E )and ?E (B ?V )χ2.

6.3.Field Star Contamination

In clusters such as NGC 2516(Fig.9),a signi?cant num-ber of foreground/background stars are present near the MS.Because the number density of these stars in each CMD in-creases towards fainter magnitudes,the distance from MS ?t-ting can be systematically overestimated.

In Figure 17we show the distribution of individual dis-tance moduli for NGC 2516from the JTH01photometry in (B ?V ,V ).Plotted are the number of stars per 0.1mag in-terval in distance modulus.The same isochrone and color ?tting ranges in Figure 9were used.In the top panel,the his-togram represents the number of stars in the catalog before photometric ?ltering.It shows the density peak that repre-sents the cluster MS,the binary sequence that extends toward shorter distance from the MS,and the ?eld star distribution that was ?tted by an exponential function (solid line ).The ?t excluded stars near the MS and the cluster binaries,and was performed over a wider range in (m ?M )0than shown here.In the middle panel the hatched histogram represents the distribution of the stars that remained after ?ltering from the catalog.This is compared with an open histogram,which was found by subtracting the exponential curve from the his-togram of the full data in the top panel.In the bottom panel we show the summed distribution of the RV members (Terndrup et al.2002)and the Chandra X-ray–detected sources (Dami-ani et al.2003)as an open histogram.The distribution of these stars after ?ltering is shown as a hatched histogram.

One of the most conspicuous features in the middle panel of Figure 17is the effective removal of cluster binaries from ?ltering.This can be seen from a de?cit of data points in the hatched histogram against the open one at shorter distances from the MS.In addition,good matches are found between these two histograms near the MS,indicating that the ?ltering algorithm has worked correctly in the foreground/background subtraction.

The RV and X-ray samples contain many cluster binaries (Fig.17,bottom panel ),but they are mostly free from ?eld star contamination.On the other hand,the full catalog data (Fig.17,top panel )are contaminated by both cluster bina-ries and foreground/background stars.Therefore,the effects of ?eld star contamination after ?ltering can be estimated by comparing distances,metallicities,and reddening values de-rived from each ?ltered set of the data (Fig.17,hatched his-tograms ).From Tables 7and 8,the weighted mean differ-ences in these quantities from the JTH01and S02photometry are ?(m ?M )S =+0.010±0.002,?[M /H]E =+0.020±0.003,?[M /H]χ2=+0.038±0.031,and ?E (B ?V )χ2=?0.003±

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聚合诸侯捍卫中原，匡正天下功业千秋。号令诸侯以匡周室，主要靠的不是 武力。 行为磊落不欺诈，美德流传于身后。孔子赞美齐桓公，也称赞管仲。 百姓深受恩惠，天子赐肉与桓公，命其无拜来接受。桓公称小白不敢，天子 威严就在咫尺前。 晋文公继承来称霸，亲身尊奉周天王。周天子赏赐丰厚，仪式隆重。 接受玉器和美酒，弓矢武士三百名。晋文公声望镇诸侯，从其风者受尊重。 威名八方全传遍，名声仅次于齐桓公。佯称周王巡狩，招其天子到河阳，因 此大众议论纷纷。 赏析 《短歌行》 (“周西伯昌”)主要是曹操向内外臣僚及天下表明心 迹，当他翦灭群凶之际，功高震主之时，正所谓“君子终日乾乾，夕惕若 厉”者，但东吴孙权却瞅准时机竟上表大说天命而称臣，意在促曹操代汉 而使其失去“挟天子以令诸侯”之号召， 故曹操机敏地认识到“ 是儿欲据吾著炉上郁!”故曹操运筹谋略而赋此《短歌行 ·周西伯 昌》。 西伯姬昌在纣朝三分天下有其二的大好形势下， 犹能奉事殷纣， 故孔子盛称 “周之德， 其可谓至德也已矣。 ”但纣王亲信崇侯虎仍不免在纣王前 还要谗毁文王，并拘系于羑里。曹操举此史实，意在表明自己正在克心效法先圣 西伯姬昌，并肯定他的所作所为，谨慎惕惧，向来无愧于献帝之所赏。 并大谈西伯姬昌、齐桓公、晋文公皆曾受命“专使征伐”。而当 今天下时势与当年的西伯、齐桓、晋文之际颇相类似，天子如命他“专使 征伐”以讨不臣，乃英明之举。但他亦效西伯之德，重齐桓之功，戒晋文 之诈。然故作谦恭之辞耳，又谁知岂无更讨封赏之意乎 ?不然建安十八年(公元 213 年)五月献帝下诏曰《册魏公九锡文》，其文曰“朕闻先王并建明德， 胙之以土，分之以民，崇其宠章，备其礼物，所以藩卫王室、左右厥世也。其在 周成，管、蔡不静，惩难念功，乃使邵康公赐齐太公履，东至于海，西至于河， 南至于穆陵，北至于无棣，五侯九伯，实得征之。 世祚太师，以表东海。爰及襄王，亦有楚人不供王职，又命晋文登为侯伯， 锡以二辂、虎贲、斧钺、禾巨 鬯、弓矢，大启南阳，世作盟主。故周室之不坏， 系二国是赖。”又“今以冀州之河东、河内、魏郡、赵国、中山、常 山，巨鹿、安平、甘陵、平原凡十郡，封君为魏公。锡君玄土，苴以白茅，爰契 尔龟。”又“加君九锡，其敬听朕命。” 观汉献帝下诏《册魏公九锡文》全篇，尽叙其功，以为其功高于伊、周，而 其奖却低于齐、晋，故赐爵赐土，又加九锡，奖励空前。但曹操被奖愈高，心内 愈忧。故曹操在曾早在五十六岁写的《让县自明本志令》中谓“或者人见 孤强盛， 又性不信天命之事， 恐私心相评， 言有不逊之志， 妄相忖度， 每用耿耿。

2008年浙师大《外国文学名著鉴赏》期末考试答案

（一）文学常识 一、古希腊罗马 1．（1）宙斯(罗马神话称为朱庇特)，希腊神话中最高的天神，掌管雷电云雨，是人和神的主宰。 （2）阿波罗，希腊神话中宙斯的儿子，主管光明、青春、音乐、诗歌等，常以手持弓箭的少年形象出现。 （3）雅典那，希腊神话中的智慧女神，雅典城邦的保护神。 （4）潘多拉，希腊神话中的第一个女人，貌美性诈。私自打开了宙斯送她的一只盒子，里面装的疾病、疯狂、罪恶、嫉妒等祸患，一齐飞出，只有希望留在盒底，人间因此充满灾难。“潘多拉的盒子”成为“祸灾的来源”的同义语。 （5）普罗米修斯，希腊神话中造福人间的神。盗取天火带到人间，并传授给人类多种手艺，触怒宙斯，被锁在高加索山崖，受神鹰啄食，是一个反抗强暴、不惜为人类牺牲一切的英雄。 （6）斯芬克司，希腊神话中的狮身女怪。常叫过路行人猜谜，猜不出即将行人杀害；后因谜底被俄底浦斯道破，即自杀。后常喻“谜”一样的人物。与埃及狮身人面像同名。 2．荷马，古希腊盲诗人。主要作品有《伊利亚特》和《奥德赛》，被称为荷马史诗。《伊利亚特》叙述十年特洛伊战争。《奥德赛》写特洛伊战争结束后，希腊英雄奥德赛历险回乡的故事。马克思称赞它“显示出永久的魅力”。 3．埃斯库罗斯，古希腊悲剧之父，代表作《被缚的普罗米修斯》。6．阿里斯托芬，古希腊“喜剧之父”代表作《阿卡奈人》。 4．索福克勒斯，古希腊重要悲剧作家，代表作《俄狄浦斯王》。5．欧里庇得斯，古希腊重要悲剧作家，代表作《美狄亚》。 二、中世纪文学 但丁，意大利人，伟大诗人，文艺复兴的先驱。恩格斯称他是“中世纪的最后一位诗人，同时又是新时代的最初一位诗人”。主要作品有叙事长诗《神曲》，由地狱、炼狱、天堂三部分组成。《神曲》以幻想形式，写但丁迷路，被人导引神游三界。在地狱中见到贪官污吏等受着惩罚，在净界中见到贪色贪财等较轻罪人，在天堂里见到殉道者等高贵的灵魂。 三、文艺复兴时期 1．薄迦丘意大利人短篇小说家，著有《十日谈》拉伯雷，法国人，著《巨人传》塞万提斯，西班牙人,著《堂?吉诃德》。 2．莎士比亚，16-17世纪文艺复兴时期英国伟大的剧作家和诗人，主要作品有四大悲剧——《哈姆雷特》、《奥赛罗》《麦克白》、《李尔王》，另有悲剧《罗密欧与朱丽叶》等，喜剧有《威尼斯商人》《第十二夜》《皆大欢喜》等，历史剧有《理查二世》、《亨利四世》等。马克思称之为“人类最伟大的戏剧天才”。 四、17世纪古典主义 9．笛福，17-18世纪英国著名小说家，被誉为“英国和欧洲小说之父”，主要作品《鲁滨逊漂流记》，是英国第一部现实主义长篇小说。10．弥尔顿，17世纪英国诗人，代表作：长诗《失乐园》，《失乐园》，表现了资产阶级清教徒的革命理想和英雄气概。 25．拉伯雷，16世纪法国作家，代表作：长篇小说《巨人传》。 26．莫里哀，法国17世纪古典主义文学最重要的作家，法国古典主义喜剧的创建者，主要作品为《伪君子》《悭吝人》（主人公叫阿巴公）等喜剧。 五、18世纪启蒙运动 1）歌德，德国文学最高成就的代表者。主要作品有书信体小说《少年维特之烦恼》，诗剧《浮士德》。 11．斯威夫特，18世纪英国作家，代表作：《格列佛游记》，以荒诞的情节讽刺了英国现实。 12．亨利·菲尔丁，18世纪英国作家，代表作：《汤姆·琼斯》。 六、19世纪浪漫主义 （1拜伦， 19世纪初期英国伟大的浪漫主义诗人，代表作为诗体小说《唐璜》通过青年贵族唐璜的种种经历，抨击欧洲反动的封建势力。《恰尔德。哈洛尔游记》 （2雨果，伟大作家，欧洲19世纪浪漫主义文学最卓越的代表。主要作品有长篇小说《巴黎圣母院》、《悲惨世界》、《笑面人》、《九三年》等。《悲惨世界》写的是失业短工冉阿让因偷吃一片面包被抓进监狱，后改名换姓，当上企业主和市长，但终不能摆脱迫害的故事。《巴黎圣母院》 弃儿伽西莫多，在一个偶然的场合被副主教克洛德.孚罗洛收养为义子，长大后有让他当上了巴黎圣母院的敲钟人。他虽然十分丑陋而且有多种残疾，心灵却异常高尚纯洁。 长年流浪街头的波希米亚姑娘拉.爱斯梅拉达，能歌善舞，天真貌美而心地淳厚。青年贫诗人尔比埃尔.甘果瓦偶然同她相遇，并在一个更偶然的场合成了她名义上的丈夫。很有名望的副教主本来一向专心于"圣职"，忽然有一天欣赏到波希米亚姑娘的歌舞，忧千方百计要把她据为己有，对她进行了种种威胁甚至陷害，同时还为此不惜玩弄卑鄙手段，去欺骗利用他的义子伽西莫多和学生甘果瓦。眼看无论如何也实现不了占有爱斯梅拉达的罪恶企图，最后竟亲手把那可爱的少女送上了绞刑架。 另一方面，伽西莫多私下也爱慕着波希米亚姑娘。她遭到陷害，被伽西莫多巧计救出，在圣母院一间密室里避难，敲钟人用十分纯朴和真诚的感情去安慰她，保护她。当她再次处于危急中时，敲钟人为了援助她，表现出非凡的英勇和机智。而当他无意中发现自己的"义父"和"恩人"远望着高挂在绞刑架上的波希米亚姑娘而发出恶魔般的狞笑时，伽西莫多立即对那个伪善者下了最后的判决，亲手把克洛德.孚罗洛从高耸入云的钟塔上推下，使他摔的粉身碎骨。 （3司汤达，批判现实主义作家。代表作《红与黑》，写的是不满封建制度的平民青年于连，千方百计向上爬，最终被送上断头台的故事。“红”是将军服色，指“入军界”的道路；“黑”是主教服色，指当神父、主教的道路。 14．雪莱，19世纪积极浪漫主义诗人，欧洲文学史上最早歌颂空想社会主义的诗人之一，主要作品为诗剧《解放了的普罗米修斯》，抒情诗《西风颂》等。 15．托马斯·哈代，19世纪英国作家，代表作：长篇小说《德伯家的苔丝》。 16．萨克雷，19世纪英国作家，代表作：《名利场》 17．盖斯凯尔夫人，19世纪英国作家，代表作：《玛丽·巴顿》。 18．夏洛蒂?勃朗特，19世纪英国女作家，代表作：长篇小说《简?爱》19艾米丽?勃朗特，19世纪英国女作家，夏洛蒂?勃朗特之妹，代表作：长篇小说《呼啸山庄》。 20．狄更斯，19世纪英国批判现实主义文学的重要代表，主要作品为长篇小说《大卫?科波菲尔》、《艰难时世》《双城记》《雾都孤儿》。21．柯南道尔，19世纪英国著名侦探小说家，代表作品侦探小说集《福尔摩斯探案》是世界上最著名的侦探小说。 七、19世纪现实主义 1、巴尔扎克，19世纪上半叶法国和欧洲批判现实主义文学的杰出代表。主要作品有《人间喜剧》，包括《高老头》、《欧也妮·葛朗台》、《贝姨》、《邦斯舅舅》等。《人间喜剧》是世界文学中规模最宏伟的创作之一，也是人类思维劳动最辉煌的成果之一。马克思称其“提供了一部法国社会特别是巴黎上流社会的卓越的现实主义历史”。

the open window 赏析

本科生课程大作业 课程名称：英国短篇小说赏 析 开课时间：2014年秋 任课教师： 学生姓名： 学生学号： 提交日期：2014-11-10

The Open Window The author is Hector Hugh Munro, whose pen name is Saki. Saki is an excellent author and playwright. The scholars always compare him to O Henry. The stories of Saki are very short, usually about five hundred. But readers enjoy a lot about his vivid plot, humorous words and extraordinary description of characters. Being similar with O Henry, Saki is famous for his bedding, the unexpected end and an amazing suspense. The Open Window is outstanding in above features. The whole story is only approximately 1,000 words while the vivid plot and highly logic structure bring us a delicious breast. Apart from his wonderful description, the subtle feeling of irony in the story is also worth analyzing. That’s also the reason I like this short story. The niece, Vera first in the story employs the ghost as the hook or suspense, which draw the attention of Mr.Nuttle immediately. Framton Nuttle is a slightly nervous person who is undergoing a nerve cure. His sister introduced him to visit Mrs. Sappleton in anther town to take a rest. The Vera actually leads the whole conversation. Then Vera tells a few words about the tragedy of Mrs. Sammpleton and takes the French window into Mr. Nuttle’s eyes. In her statement, she emphasizes the tragedy just 3 years ago , which is because that Mr. Nuttle’s aunt lived there 4 years ago. The hook is so great and captivated. In England, an October day was a little cold especially in the evening, nearly below 0 Celsius. But the window is still open. It is abnormal for Mr.Nuttle. The whole suspense is so successful for the Mr.Nuttle who is curious about the open window and asks the reason. Next, Vera has the chance to continue her lie and tell Mr.Nuttle that Mrs Sammpleton believes that her husband and her brothers, who were killed in a shooting accident three years before, will come back one day. In this sense, if her husband and brothers are back, they must be the ghosts. To make Vera herself more trustful, Vera shows her excellent play. When Mrs Sammpleton points out that her husband and her brothers are coming, Mr.Nuttle sees the facial look of Vera and look out of the window. Vera “was staring out through the open window with a dazed horror in her eyes”. The facial look of Vera is appealing and creating a horror atmosphere. There must be something unusual thing happening. To Mr.Nuttle’s horror, he thinks he is seeing ghosts and running away. Vera employs all the normal elements to establish the ghost story which successfully make the nervous Mr.Nuttle get away. But the end is different from the ghost story and also reasoning in logic. There is no ghost actually. It is only the tricks by Vera. And it is the ghosts in Mr.Nuttle’s mind that tease himself. However, the short novel is more than telling a wonderful story. The unnecessary and over-elaborate formalities in society is also an irony. On this point, we need to focus back to Saki’s life experience. Saki was brought up by his two aunts who flatter their status and emphasize on rituals only, without loving and mercy actually. In their mind, being a good manner, obeying the law and to be polite kid is better than one with fair sole, mercy, loving and tolerance. Undergoing the parenting by aunts, Saki had experienced a bored childhood. On this

外国名著赏析论文

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